ON POSITIVITY OF THE GREEN FUNCTION FOR POISSON PROBLEM FOR A LINEAR FUNCTIONAL DIFFERENTIAL EQUATION

Abstract

For the Poisson problem -∆u+p x u- Ωu s r x, ds=ρf, u | Γ( Ω )=0 equivalence of positivity of the Green function and other classical properties is showed. Here Ω is an open set in R n , and Γ( Ω ) is the boundary of the Ω . For almost all x∈ Ω , r(x, ∙) is a measure satisfying certain symmetry condition. In particular this equation involves integral differential equation and the equation -∆u+p x u(x)- i=1 m p i x u h i x =ρf, where h i : Ω→Ω is a measurable mapping.